Finite Group Databasebeta

← #28600895

Subgroups — #28600895

Order 15104.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426845C211yesmaximal, characteristic, complemented
2#21426739C211yesmaximal, characteristic
3#21425049C211yesmaximal, characteristic, complemented
4#21424866C211yesmaximal, characteristic, complemented
5#21424840C211yesmaximal, characteristic, complemented
6#21424825C211yesmaximal, characteristic, complemented
7#21424795C211yesmaximal, characteristic
8#21424781C211yesmaximal, characteristic
9#21424725C211yesmaximal, characteristic
10#21424718C211yesmaximal, characteristic
11#21424683C211yesmaximal, characteristic, complemented
12#21424676C211yesmaximal, characteristic, complemented
13#21424106C211yesmaximal, characteristic
14#21423955C211yesmaximal, characteristic, complemented
15C2×C59×#476C211yesmaximal, characteristic
16C2×#14273661C2211yescharacteristic
17C2×#14273659C2211yescharacteristic
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19C2×#14273648C2211yescharacteristic
20C2×#14273648C2211yescharacteristic
21C2×#14273637C2211yescharacteristic
22C2×C8×Dic59C2211yescharacteristic
23#16837309—21no—
24#16837309—21no—
25#16837305C2211yescharacteristic
26#16837305C2211yescharacteristic
27#16837301C2211yescharacteristic
28#16837301C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837301C2211yescharacteristic
31#16837295—21no—
32#16837295—21no—
33#16837288—21no—
34#16837288—21no—
35#16837268C2211yescharacteristic
36#16837265—21no—
37#16837262—21no—
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41#16837248C2211yescharacteristic
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×C4⋊Q8×C59C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C59×#476C418yescomplemented
55C2×#106×C59C2211yescharacteristic
56C2×#106×C59C2211yescharacteristic
57C2×C4⋊C8×C59C2211yescharacteristic
58C2×#102×C59C2211yescharacteristic
59C2×#102×C59C2211yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C2×C4⋊Dic59—21no—
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C22⋊Dic59—21no—
71C2×C22⋊Dic59—21no—
72C2×C22⋊Dic59—21no—
73C2×C22⋊Dic59—21no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59C2311yescharacteristic
78C2×C4×Dic59C2311yescharacteristic
79#14273661—41no—
80#14273659—41no—
81C4⋊(C59⋊C8)—22no—
82#14273648—41no—
83#14273648—41no—
84#14273637—41no—
85C8×Dic59—41no—
86C59⋊#95—21no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95D411yescharacteristic
96C59⋊#95D411yescharacteristic
97C59⋊#95D411yescharacteristic
98C59⋊#95D411yescharacteristic
99C22×Q8×C59—21no—
100C22×D4×C59C2311yescharacteristic
101C23×C4×C59—21no—
102C22×C8×C59C2311yescharacteristic
103C22×C8×C59C2311yescharacteristic
104C4⋊Q8×C59—22no—
105C4⋊Q8×C59C2×C414yes—
106C4×D4×C59—22no—
107C4×D4×C59C2×C414yes—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C22⋊C4×C59—21no—
113C2×C42×C59C2311yescharacteristic
114#106×C59C2×C414yes—
115#106×C59C2×C414yes—
116C4⋊C8×C59C2×C414yes—
117#102×C59C2×C414yes—
118#102×C59C2×C414yes—
119C24×C59—21no—
120C2×Q8×C59—24no—
121C2×Q8×C59—81no—
122C2×Q8×C59—81no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59C22×C412yes—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—24no—
133C22×C4×C59—81no—
134C22×C4×C59—81no—
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59C2×D411yescharacteristic
143C22×Dic59C2×D411yescharacteristic
144C22×Dic59C2×D411yescharacteristic
145C22×Dic59C2×D411yescharacteristic
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59C22×C412yes—
151C2×C8×C59C22×C412yes—
152C4⋊C4×C59—24no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59C22×C412yes—
157C4⋊C4×C59C22×C412yes—
158C4⋊C4×C59C22×C412yes—
159C22⋊C4×C59—24no—
160C42×C59C22×C412yes—
161C42×C59#4112yes—
162C22⋊Dic59—81no—
163C22⋊Dic59—81no—
164C22⋊Dic59—81no—
165C22⋊Dic59—81no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—81no—
175C4⋊Dic59—81no—
176C4⋊Dic59—81no—
177C4⋊Dic59—81no—
178C4×Dic59—21no—
179C4×Dic59—21no—
180C4×Dic59—21no—
181C4×Dic59—21no—
182C4×Dic59—81no—
183C4×Dic59—81no—
184C4×Dic59—81no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C23×C59—22no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59C22×D411yescharacteristic
199Q8×C59—84no—
200D4×C59—22no—
201D4×C59—22no—
202D4×C59—41no—
203D4×C59—41no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—22no—
209C2×C4×C59—22no—
210C2×C4×C59—22no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—84no—
218C2×C4×C59C4×D412yes—
219C2×C4×C59C4×D412yes—
220C2×C4×C59C23×C411yescharacteristic
221C2×C4×C59C2×#4111yescharacteristic
222C2×C4×C59H(5,𝔽2)11yescharacteristic
223C2×C4×C592−1+411yescharacteristic
224C2×C4×C592−1+411yescharacteristic
225C2×C4×C592−1+411yescharacteristic
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C8×C59—22no—
251C8×C59—22no—
252C59⋊C8—41no—
253C59⋊C8—41no—
254#22123—591nomaximal, Sylow
255C22×C59—42no—
256C22×C59—42no—
257C22×C59—42no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59C2×C4×D411yescharacteristic
263C22×C59#53611yescharacteristic
264C22×C59#53711yescharacteristic
265C22×C59#57611yescharacteristic
266C22×C59#57711yescharacteristic
267C22×C59#57711yescharacteristic
268C22×C59#57711yescharacteristic
269C4×C59—22no—
270C4×C59—22no—
271C4×C59—42no—
272C4×C59—42no—
273C4×C59—42no—
274C4×C59#51911yescharacteristic
275C4×C59#52011yescharacteristic
276C4×C59#54111yescharacteristic
277C4×C59#54211yescharacteristic
278Dic59—41no—
279Dic59—41no—
280Dic59—41no—
281Dic59—41no—
282Dic59—81no—
283Dic59—81no—
284Dic59—81no—
285Dic59—81no—
286C2×#476D5911yescharacteristic
287#2331—591no—
288#2319—591no—
289#1988—591no—
290#1949—591no—
291#1944—591no—
292#1941—591no—
293#1936—591no—
294#1934—591no—
295#1926—591no—
296#1925—591no—
297#1920—591no—
298#1919—591no—
299#1804—591no—
300#1768—591no—
301C2×C59—82no—
302C2×C59#232311yescharacteristic
303C2×C59#294611yescharacteristic
304C2×C59#294911yescharacteristic
305C2×C59#324111yescharacteristic
306C2×C59#324411yescharacteristic
307C2×C59#325711yescharacteristic
308C2×C59#326311yescharacteristic
309C2×C4⋊Q8C2×D5911yescharacteristic
310C2×C4×D4C2×D5911yescharacteristic
311#476Dic5918yescomplemented
312C2×#107—591no—
313C2×#106—591no—
314C2×#106C2×D5911yescharacteristic
315C2×#106C2×D5911yescharacteristic
316C2×C4⋊C8—591no—
317C2×C4⋊C8C2×D5911yescharacteristic
318C2×#102—591no—
319C2×#102—591no—
320C2×#102C2×D5911yescharacteristic
321C2×#102C2×D5911yescharacteristic
322C2×#97—591no—
323C2×C4×C8—591no—
324#391—1181no—
325#391—1181no—
326#390—591no—
327#390—591no—
328#389—591no—
329#389—591no—
330#389—591no—
331#389—591no—
332#388—1181no—
333#388—1181no—
334#387—1181no—
335#387—1181no—
336#382—591no—
337#381—1181no—
338#380—1181no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4—591no—
341C4×C4⋊C4—591no—
342#376—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#336—591no—
352#336—591no—
353C59#2212311yescharacteristic, complemented, Sylow
354C22×Q8—21no—
355C22×D4C22×D5911yescharacteristic
356C23×C4—21no—
357C22×C8—591no—
358C22×C8—591no—
359C22×C8C22×D5911yescharacteristic
360C22×C8C22×D5911yescharacteristic
361C4⋊Q8—22no—
362C4⋊Q8C2×Dic5914yes—
363C4×D4—22no—
364C4×D4C2×Dic5914yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C4⋊C4C22×D5911yescharacteristic
376C2×C4⋊C4C22×D5911yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C22⋊C4—1181no—
382C2×C42—591no—
383C2×C42—591no—
384C2×C42—1181no—
385C2×C42—1181no—
386C2×C42—1181no—
387C2×C42C22×D5911yescharacteristic
388#107—2361no—
389#106—2361no—
390#106C2×Dic5914yes—
391#106C2×Dic5914yes—
392C4⋊C8—1182no—
393C4⋊C8C2×Dic5914yes—
394#102—2361no—
395#102—2361no—
396#102C2×Dic5914yes—
397#102C2×Dic5914yes—
398#97—2361no—
399C4×C8—2361no—
400#95—591no—
401#95—591no—
402#95—591no—
403#95—591no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413C24—21no—
414C2×Q8—24no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C22×Dic5912yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—591no—
430C22×C4—591no—
431C22×C4—591no—
432C22×C4—591no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4D4×D5911yescharacteristic
438C22×C4D4×D5911yescharacteristic
439C22×C4C23×D5911yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8C22×Dic5912yes—
451C2×C8C22×Dic5912yes—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—4721no—
465C4⋊C4—4721no—
466C4⋊C4—4721no—
467C4⋊C4—4721no—
468C4⋊C4C22×Dic5912yes—
469C4⋊C4C22×Dic5912yes—
470C4⋊C4C22×Dic5912yes—
471C22⋊C4—24no—
472C22⋊C4—4721no—
473C22⋊C4—4721no—
474C22⋊C4—4721no—
475C22⋊C4—4721no—
476C42—1181no—
477C42—1181no—
478C42—1181no—
479C42—1181no—
480C42—4721no—
481C42—4721no—
482C42—4721no—
483C42#1174568712yes—
484C42C22×Dic5912yes—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C2×D4×D5911yescharacteristic
493Q8—84no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4—1181no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1181no—
516C2×C4—1181no—
517C2×C4—1181no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4D4×Dic5912yes—
537C2×C4D4×Dic5912yes—
538C2×C4C23×Dic5911yescharacteristic
539C2×C4C2×#1174568711yescharacteristic
540C2×C4#1427377311yescharacteristic
541C2×C4#1427377411yescharacteristic
542C2×C4#1427377511yescharacteristic
543C2×C4#1427377511yescharacteristic
544C8—22no—
545C8—22no—
546C8—2361no—
547C8—2361no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22C2×D4×Dic5911yescharacteristic
556C22#1683797511yescharacteristic
557C22#1683798211yescharacteristic
558C22#1683820611yescharacteristic
559C22#1683820811yescharacteristic
560C22#1683820811yescharacteristic
561C22#1683821211yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—4721no—
568C4—4721no—
569C4—4721no—
570C4—4721no—
571C4—2361no—
572C4—2361no—
573C4—2361no—
574C4—2361no—
575C4#1683788011yescharacteristic
576C4#1683788511yescharacteristic
577C4#1683801211yescharacteristic
578C4#1683802511yescharacteristic
579C2—82no—
580C2#2142678011yescharacteristic
581C2#2143292911yescharacteristic
582C2#2143295611yescharacteristic
583C2#2143534811yescharacteristic
584C2#2143539111yescharacteristic
585C2#2143550611yescharacteristic
586C2#2143556811yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#476C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenterC2×#476D59
hypocenterC59#22123
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21426845C2semidirect
3. #21425049C2semidirect
4. #21424866C2semidirect
5. #21424840C2semidirect
6. #21424825C2semidirect
11. #21424683C2semidirect
12. #21424676C2semidirect
14. #21423955C2semidirect
54. C59×#476C4semidirect
311. #476Dic59semidirect
353. C59#22123semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214268452
#214267392
#214250492
#214248662
#214248402
#214248252
#214247952
#214247812
#214247252
#214247182
#214246832
#214246762
#214241062
#214239552
C2×C59×#4762
#2212359